Is the sum of 13 riyals owed to a brother considered a debt that must be repaid, and how should this debt be handled after many years have passed and the brother no longer owns the commercial store, especially given the awkwardness of informing him about it?

This amount is a debt owed to your brother, and it must be repaid to him unless he forgives you. It is not permissible to donate it as long as you are able to deliver it to him. Your feeling of embarrassment does not exempt you from the obligation to repay. You can deliver the money to him by any suitable means without informing him that it is a debt.